# What is the relationship between a pair of relatively prime integers and two consecutive integers?

Suppose we have two integers $$b$$ and $$c$$.

If $$\gcd(b, c) = 1$$, then the pair of integers is relatively prime.

$$\gcd(b, c) = 1$$ can also be expressed as a linear combination $$bx_0 + cy_0 =1$$, where $$x_0$$ and $$y_0$$ are integers chosen such that $$bx_0 + cy_0$$ is the smallest positive element in the set $$\{bx + cy\}$$, where $$x$$ and $$y$$ are integers.

From the above, we can conclude that all pairs of consecutive integers are relatively prime pairs considering:

$$(1)(x+1) + (-1)(x) = 1 \implies \gcd(x, x+1) = 1$$ for an integer $$x$$

Now, consider the relatively prime pair of numbers $$37$$ and $$39$$. Because $$\gcd(37, 39) = 1$$ we know that there must be some $$x_0$$ and some $$y_0$$ for which $$37x_0 + 39y_0 = 1$$. That means there are two consecutive integers represented by $$37x_0$$ and $$39y_0$$. (We also know that a pair of consecutive integers must correspond to every pair of relatively prime numbers that is not consecutive because of equality of the linear combination to 1).

I have found $$x_0 = 19, y_0 = 18$$ representing the pair of consecutive numbers $$\gcd(703, 702)$$ that is related to $$\gcd(37, 39)$$.

I want to better understand the relationship between pairs of relatively prime numbers that are not consecutive numbers and their relatively prime numbers that are consecutive numbers. For example, is there a unique pair of consecutive numbers for every relatively prime set of nonconsecutive numbers?

$$(6)+(-5) = (2){\color{red}{(3)}}+(-1){\color{red}{(5)}}=1$$ $$(-9)+(10) = (-3){\color{red}{(3)}}+(2){\color{red}{(5)}}=1$$

For $$(a,b)=1$$, then there exist integers $$m,n$$ with $$ma+nb=1$$.

By adding and subtracting $$ab$$, we get a new solution

$$ma+nb +ab - ab = 1 \iff (m+b)a + (n-a)b = 1$$

You can continue this indefinitely.

And so one question you may ask is:

How often can the sequence $$\{(|(m+kb)a|, |(n-ka)b|)\}_{k\in\mathbb Z}$$ be a consecutive pair?

• Would there be a way to know exactly how many pairs of consecutive numbers are associated with any pair of nonconsecutive numbers? Dec 1, 2019 at 1:39
• Hopefully my edit helps you some Dec 1, 2019 at 2:48